The van der Waerden Number W (2, 6) Is 1132
Michal Kouril and Jerome L. Paul
CONTENTS
1. Introduction
2. Preprocessing: Eliminating Redundancies
3. Finding Unavoidable Patterns
4. Preprocessing Patterns: Removing Redundant Branches
5. The Palindrome Heuristic
6. Putting the Preprocessing Together for W (2, 6)
7. SAT Solver for van der Warden K = 2 Numbers
8. Architecture of the FPGA-Based SAT Solver
9. Performance Results
10. Extreme Partitions
11. Conclusion and Future Work
Acknowledgments
References
We have verified that the van der Waerden number W (2, 6) is
1132, that is, 1132 is the smallest integer n = W (2, 6) such that
whenever the set of integers {1, 2, . . . , n} is 2-colored, there exists a monochromatic arithmetic progression of length 6. This
was accomplished by applying special preprocessing techniques
that drastically reduced the required search space. The exhaustive search showing that W (2, 6) = 1132 was carried out by
formulating the problem as a satisfiability (SAT) question for a
Boolean formula in conjunctive normal form (CNF), and then
using a SAT solver specifically designed for the problem. The
parallel backtracking computation was run over multiple Beowulf clusters, and in the last phase, field programmable gate
arrays (FPGAs) were used to speed up the search. The fact that
W (2, 6) > 1131 was shown previously by the first author.
1.
2000 AMS Subject Classification: Primary 68R05, 05D10
Keywords: Van der Waerden numbers, combinatorics,
high-performance computing, Beowulf clusters, FPGAs
INTRODUCTION
In 1926, B. L. van der Waerden proved the following result [van der Waerden 27]: given positive integers K,
L, both at least 2 (the case of 1 being trivial), there is a
smallest integer n = W (K, L) such that every K-coloring
of {1, 2, . . . , n} contains a monochromatic arithmetic progression of length L. Van der Waerden’s proof gave little
insight into the actual value of W (K, L), and much interest has been shown in the combinatorial community
in finding actual values for these numbers, or improving
known lower bounds for them. In this paper, we restrict
attention to K = 2, where the only nontrivial values
heretofore known were W (2, 3) = 9, W (2, 4) = 35, and
W (2, 5) = 178. This last value was computed by Beeler
and O’Neil in 1979 [Beeler and O’Neil 79].
In [Kouril and Franco 05] it was shown that W (2, 6) >
1131 by exhibiting 2-colorings of {1, 2, . . . , 1131} not containing monochromatic arithmetic progressions of length
6 (see Section 8 of the current paper for a description of these 2-colorings). In this paper we show that
W (2, 6) = 1132. In order to carry out the massive computation, unavoidable patterns were utilized, and equivalent (and therefore redundant) branches in the search tree
were pruned. For given integers K, L, and n < W (K, L),
c A K Peters, Ltd.
1058-6458/2008 $ 0.50 per page
Experimental Mathematics 17:1, page 53
54
Experimental Mathematics, Vol. 17 (2008), No. 1
FIGURE 1. Number of partitions per partition length.
we say that a pattern P is (K, L, n)-unavoidable if P
(or its various equivalent forms; see below) must exist in any K-coloring of {1, 2, . . . , n} not containing an
arithmetic progression of length L. Note that if P is
(K, L, n)-unavoidable, then it is (K, L, m)-unavoidable
for all n ≤ m < W (K, L).
The following were the main steps in determining that
W (2, 6) = 1132:
1. Show that the pattern 0000 (equivalently, 1111)
is (2, 6, 240)-unavoidable.
In other words, any
2-coloring of {1, 2, . . . , 240} not containing a
monochromatic arithmetic progression of length 6
will contain either 0000 or 1111 (and hence, for example, must also contain either 00001 or 11110).
2. Start with an unavoidable pattern 00001 and grow
the patterns while removing the redundant branches
using a so-called palindrome heuristic. Do so while
the set is manageable. In our case we stop at patterns of length at most 29, which yields 2,537,546
initial patterns.
3. Take the patterns generated in step 2, and convert
(in the obvious way) the problem of extending them
to patterns without monochromatic arithmetic progressions of length 6 to the problem of solving a set of
2,537,546 SAT problems. Here the associated SAT
problem is satisfiable if the variables can be 2-colored
without a monochromatic arithmetic progression of
length 6 occurring in the corresponding pattern (we
say that such a pattern is satisfying). The total
number of variables in our set of SAT problems was
taken as 240, and we placed the 2,537,546 patterns
of length 28 or 29 found in step 2 (preset variables)
in the middle portion of {1, 2, . . . , 240} for extending
to satisfying patterns of length 240.
4. Solve each SAT problem from step 3. It turns out
that only 111 out of 2,537,546 were found satisfiable.
We used Beowulf clusters and later FPGA circuits
to process all of the SAT problems.
5. Collect all solutions (satisfying patterns) for all satisfiable SAT problems from step 4.
6. To reduce the number of satisfying patterns further,
we added 86 more unknown variables, bringing the
total to 326 variables. Only 52 of the original 111
patterns found in step 4 can be extended to satisfying patterns of length 326, and there are 648,005
such patterns.
7. In order to re-create the original search space
that was compressed in steps 1 and 2, shift all
patterns from step 6 to all possible positions in
{1, 2, . . . , 1131} and expand to satisfying patterns of
length 1131. This yielded 3552 nonidentical satisfying patterns.
8. None of the resulting 3552 satisfying patterns of
length 1131 from step 7 were extendable without introducing a monochromatic arithmetic progression
of length 6, and therefore W (2, 6) = 1132.
Kouril and Paul: The van der Waerden Number W (2, 6) Is 1132
55
FIGURE 2. Number of 2-colored partitions without monochromatic arithmetic progressions of length 5.
Remark 1.1. During the analysis [Kouril 06] of W (2, 3),
W (2, 4), and W (2, 5), the following property was observed: the number of all possible 2-colored patterns
without an arithmetic progression of length L (for 3, 4,
and 5) grows exponentially in the beginning, eventually
hits a peak, and then falls exponentially (see Figure 2).
It turns out that as L increases, the peak shifts to the left
as a function of the length of the partition measured by
a percentage of W (2, L) (see Figure 1). The dip on the
other side of the peak eventually levels and starts slowly
growing again. The lowest point in the dip indicates that
there is only a fairly limited number of satisfying patterns
of length m even though m is much smaller than W (2, L).
Our approach was partially based on this observation. By
collecting all possible patterns near but past the bottom
of the dip (in our case of W (2, 6) we took m = 240, later
improved to 326) and extending them to their maximum
satisfying length, it became feasible to compute W (2, 6).
2.
PREPROCESSING: ELIMINATING REDUNDANCIES
During the preprocessing, some of the branches will not
be searched, because they are found to be equivalent to
other branches. There are two operations on a pattern,
namely negation and reversal, that yield equivalent patterns in the sense that the pattern can be extended to
a satisfying pattern of a given length if and only if the
patterns transformed by either one or both (or neither)
of the operations can be so extended to a pattern of the
same length. More formally, we consider the following
functions:
• negation(p) returns the same pattern with the colors
interchanged (negated).
Example: negation(00001) = 11110.
• reverse(p) returns the same pattern but in reverse.
Example: reverse(00001) = 10000.
Two patterns are said to be equivalent if one can be
transformed to the other by applying one or both (or neither) of these two functions. For example, 001 is equivalent to 110, 100, and 011. For a pattern p, we define
minimal(p) to be the equivalent pattern having the smallest binary value. For example, minimal(1011) = 0010.
Obviously, a pattern p contains an arithmetic progression of length L if and only if minimal(p) contains such a
progression. As part of our pattern-expansion technique,
this will allow for significant pruning of the search tree.
Note that two patterns p1 , p2 are equivalent if and only
if minimal(p1 ) = minimal(p2 ).
We recognize two types of pattern palindromes:
(a) positive palindromes (identical after reversal);
Examples: 00100, 1001.
(b) negative palindromes (identical after reversal and
negation; requires n to be even);
56
Experimental Mathematics, Vol. 17 (2008), No. 1
K
2
2
2
2
2
2
2
2
2
2
L
L
L
4
4
5
5
5
6
6
6
Pattern
01/10
00/11
00/11
000/111
00/11
000/111
0000/1111
00/11
000/111
0000/1111
m
L−1
2L − 1
7
21
9
61
85
11
190
240
TABLE 1. Minimal m for various unavoidable patterns.
and two unassigned) to a given pattern. This notion of
an extension allows for an efficient identification of equivalent branches in the search tree.
To demonstrate the identification of equivalent
branches in the search space, we start with a pattern
of a single element 0 and extend on the right, with x
denoting an unassigned variable:
x00x
x01x
Extend on the right again:
xx000xx
Examples: 1100, 101010.
xx001xx
Palindromes will play a critical role in eliminating redundancies.
xx010xx
xx011xx
3.
FINDING UNAVOIDABLE PATTERNS
It is sometimes easy to find (K, L, m)-unavoidable patterns. For example, 01 is a (K, L, L)-unavoidable pattern
for L > 2. Also, 001 is a (2, L, 2L − 1)-unavoidable pattern for L > 2. Finding m for longer patterns becomes
increasingly difficult. Using a computer search we established m for selected patterns. The longest pattern found
for K = 2 and L = 6 took several weeks on a single computer. We used a modified SAT solver that backtracked
not only when it found a monochromatic arithmetic progression, but also when it found one of the unavoidable
patterns. In Table 1, we show the minimal m for which
(2, L, m)-unavoidable patterns exist.
4.
PREPROCESSING PATTERNS: REMOVING
REDUNDANT BRANCHES
In exploring the search space, the patterns resulting from
the extension of a suitable set of minimal patterns of
length m will cover all patterns of length n = n(m) in
the sense that all 2-colorings of {1, 2, . . . , n} will contain
one of the patterns of length m or their equivalents. In
our case the number of extensions to the initial pattern
did not increase the covered n beyond the lower bound
of W (2, 6).
Given a pattern P , an expansion of P consists first in
adding a zero or one to the right or left of P , resulting in
what we call an augmented pattern P . Then we add two
unassigned padding variables to the left and to the right
of this P . The addition of the two padding variables
allows the expanded pattern to be reversed and still contain the reversal of the augmented pattern. Note that
each expansion adds three new variables (one assigned
Note that the fourth pattern is equivalent to the second by a reversal and negation, so it can be eliminated,
and the three remaining patterns will cover the search
space of all patterns of length 7. Here we are not implying that the resulting patterns will be exactly at the
middle position. We are merely saying that they will
be somewhere in the covered search space with n ≥ 7.
To recover the entire search space we need to shift these
patterns and their variations over the entire {1, 2, . . . , n}
using negation and reversal. If we consider patterns only
in their minimal form and eliminate redundancies assuming an initial pattern length i and m expansions, the resulting set of patterns will cover all patterns of length
n = i + 3 × m.
Now assume that there is a sufficient number of unassigned variables on each side and focus only on the expanded patterns. Starting from a single 0, we have a
choice to expand this pattern either on the right (a) or
on the left (b), leading to the following patterns in minimal form:
(a) 0x → 00, 01
(b) x0 → 00, 10 → 00, 01
Notice that no matter whether you grow the patterns
on the left or on the right, the resulting sets of patterns
in minimal form are the same. Notice also that the two
patterns generated are palindromes. It is easy to see
that the same set of minimal patterns results whether a
palindrome is expanded on the left or on the right. To
illustrate, we go to the next step:
(a) 00x, 01x → 000, 001, 010, 011 → 000, 001, 010, 001
Kouril and Paul: The van der Waerden Number W (2, 6) Is 1132
57
(b) 00x, x01 → 000, 001, 001, 101 → 000, 001, 001, 010
(c) x00, 01x → 000, 100, 010, 011 → 000, 001, 010, 001
(d) x00, x01 → 000, 100, 001, 101 → 000, 001, 001, 010
Each set consists of 000, 001, 010. Going yet another
step yields
(a) 000x, 001x, 010x → 0000, 0001, 0010, 0011, 0100, 0101
FIGURE 3. Tree of minimal patterns for four numbers.
(b) 000x, 001x, x010 → 0000, 0001, 0010, 0011, 0010, 1010
(c) 000x, x001, 010x → 0000, 0001, 0001, 1001, 0100, 0101
(d) 000x, x001, x010 → 0000, 0001, 0001, 1001, 0010, 1010
(e) x000, 001x, 010x → 0000, 1000, 0010, 0011, 0100, 0101
(f) x000, 001x, x010 → 0000, 1000, 0010, 0011, 0010, 1010
(g) x000, x001, 010x → 0000, 1000, 0001, 1001, 0100, 0101
(h) x000, x001, x010 → 0000, 1000, 0001, 1001, 0010, 1010
After the transformation we get two different sets of
patterns:
(i) 0000, 0001, 0010, 0011, 0101
FIGURE 4. Tree of minimal patterns for six numbers.
(ii) 0000, 0001, 0010, 0110, 0101
In general, for every existing pattern we have a choice
of extending it either on the left or on the right. The
number of patterns to consider can be greatly reduced
by a careful choice of the direction in which they are extended, since some of them extend to identical patterns.
Finding the optimal way to extend patterns is a hard
problem, and looking for the best solution by enumerating all possibilities yielded results for only a few steps.
Therefore we came up with a palindrome-based heuristic
discussed in the next section.
Figures 3 and 4 show pattern-expansion trees. In Figure 3 the only pattern in which the choice of the direction of expansion matters is 001. It expands either to two
patterns to the lower left or to two patterns to the lower
right. The other patterns are palindromes, and it never
matters for such patterns whether they are expanded on
the left or on the right.
Highlighted patterns in Figure 4 indicate which patterns have a choice of expansion on the left or on the
right. Unhighlighted patterns are palindromes, which
yield the same patterns if expanded on the left or on the
right. It is obvious that the number of patterns for which
the choice matters increases significantly as the portion
of palindromes drops.
5.
THE PALINDROME HEURISTIC
The best approximation to pattern extension we have
seen so far is based on palindrome patterns, whereby an
extension on each end of such a pattern yields three new
patterns instead of four.
The following is an example of the expansion process:
x010x
00100 → 00100
10101 → 01010
00101 → 00101
10100 → 00101
Since the last two patterns are equivalent, one of them
can be eliminated.
Here is another example:
x0011x
000111 → 000111
100110 → 011001
000110 → 000110
100111 → 000110
Again, last two patterns are equivalent, and one of
them can be eliminated. Another advantage is that two
of the newly formed patterns are again palindromes.
58
Experimental Mathematics, Vol. 17 (2008), No. 1
Iteration
Palindromes
Nonpalindromes
0
1
2
3
4
5
6
7
8
9
10
11
12
0
2
4
8
14
28
46
97
176
363
722
1218
2429
1
0
3
18
78
319
1252
4581
17034
62092
224113
763984
2535117
Maximum
Length
5
7
9
11
13
15
17
19
21
23
25
27
29
Total Count
1
2
7
26
92
347
1298
4678
17210
62455
224835
765202
2537546
TABLE 2. Counts of patterns during palindrome-based pattern growing.
The nonpalindromes are extended so that they reach
a palindrome in the shortest number of extension steps.
For example, 0001 is not a palindrome, but if extended
on the left it yields a palindrome (10001) and a nonpalindrome (00001).
As demonstrated earlier, when starting with 00001,
after one expansion we have safely covered pattern length
n ≥ 8; after the second expansion we have safely covered
pattern length n ≥ 11; and so forth. Since in our case we
are doing at most 24 expansions with the initial pattern
of length 5, we would safely cover pattern length n ≥ 77.
In addition, our initial pattern 00001 is unavoidable
for n > 239. We assume that it can appear anywhere in
patterns of length 240 or more. To account for all possible
positions we will have to shift the resulting patterns (after
crossing the peak; see Remark 1.1) within certain pattern
lengths. A safe estimation of this pattern length is based
on the fact that we can place the initial pattern at either
side of a pattern of length 240 and add the covered search
space from the subsequent expansion. Therefore n >
239 + 77 + 77 = 393. This can undoubtedly be tightened,
but with the lower bound for W (2, 6) being 1131, this
will suffice.
To recover the original search space we have only to
shift and extend the final set of patterns throughout
{1, 2, . . . , n}. The shifting–extending phase of the patterns is computationally feasible, since the length of the
patterns in our set is sufficient past the peak [Kouril and
Franco 05].
6.
PUTTING THE PREPROCESSING TOGETHER
FOR W (2, 6)
The longest unavoidable pattern we have verified for
W (2, 6) is 0000. This pattern can even be extended to
00001 without loss of generality. Growing this pattern
when the lower bound is 1131 gives us plenty of space for
a number of iterations of pattern expansion. We grow the
initial pattern while the number of patterns is manageable, stopping after the twelfth iteration when the patterns are of length 28 or 29, and result in a total count
of 2,537,546 patterns (see Table 2):
1. 00001
2. 100001, 000001
3. 01000010, 010000010, 00111100, 001111100,
001111010, 001111011, 001111101
4. . . .
To show the dramatic effect of the palindrome heuristic, there is a total of 155,896,884 patterns of length
29 without a monochromatic arithmetic progression of
length 6, versus the 2,537,546 found using the heuristic.
7.
SAT SOLVER FOR VAN DER WAERDEN K = 2
NUMBERS
For convenience we transform the problem of evaluating
van der Waerden numbers into an equivalent problem of
satisfiability of a certain Boolean expression in conjunctive normal form (CNF). With each integer i in the vector
{1, 2, . . . , n} we associate a Boolean variable xi , and to
each 2-coloring of {1, 2, . . . , n} we assign xi true or false,
one value for each color. The CNF associated with a
2-coloring simply enumerates all possible progressions in
order to cause a contradiction whenever it occurs. For
example, (x1 ∨x2 ∨x3 ∨x4 ∨x5 ∨x6 ) will cause a contradiction whenever the variables x1 , x2 , x3 , x4 , x5 , and x6 are
assigned false, which is also an arithmetic progression of
Kouril and Paul: The van der Waerden Number W (2, 6) Is 1132
length 6 on positions 1 to 6. Our solver does not store the
problem explicitly in CNF, but implicitly does checking
for progressions as though it were stored in CNF.
We tried various algorithms to find the fastest SAT
solver that would answer the question whether a given
partial assignment can be extended to an assignment of
the remaining unknown variables such that there is no
monochromatic arithmetic progression of length L. It
turned out that the fastest algorithm we found was a
DPLL SAT solver restricted to checking for inferences
and contradictions. The single function that dominates
all other computations checks whether the newly assigned
variable is part of a progression (in which case it triggers
a contradiction) or is part of an L − 1 progression with
an unknown variable present (in which case it triggers
an inference). We eventually used FPGAs to do this
checking, which resulted in considerable speedup.
Similar attention for the sake of efficiency was given to
finding the best branching heuristic. None of the tested
special heuristics yielded any faster result than the static
one that began selecting variables on which to branch
from the middle outward. If the selected variable was
already assigned, it was skipped, and the next one was
tried.
8.
ARCHITECTURE OF THE FPGA-BASED SAT SOLVER
As mentioned above, our solver searched the search space
of 2-colorings of {1, 2, . . . , n} for a coloring not containing a monochromatic arithmetic progression of a certain
length. Each number has a color, for which we have two
registers of length n: C0 and C1 , one for each color. At
the start of the computation, 0 is assigned to both registers for all numbers except those that are part of the
initial assignment. An assignment of 1 to either register
constitutes a coloring of that number with the respective color. Assigning both registers the value 1 is not a
possible state.
Our design consists of the following major blocks (see
Figure 5):
1. IB: Inference block
FIGURE 5. The FPGA solver block diagram.
FIGURE 6. Contradiction-detection block.
The CD (contradiction-detection block) has a color
vector as input, and the output is true or false depending on whether the input vector contains an arithmetic
progression.
For example, if the input is 01111110001, the output
will be 1 (true), since positions 2 to 7 contain an arithmetic progression of length 6.
The IB (inference block; see Figure 7) has a color vector as input, and the output is a list of inferences, i.e., a
list of colorations forced to occur in the other color.
For example, if the input vector is 111101, it is obvious
that we cannot color the fifth position using the same
color, and therefore the opposite color is inferred. The
output of the IB is 000010.
The CP (choice-point-selection block; see Figure 8)
has an input consisting of both color vectors and an output length n. The output will have only one bit set, which
corresponds to the next unassigned number. Our solver
uses a simple selection of the next choice point by selecting the numbers to color starting from the middle and
working its way out. The CP selects the next unassigned
number that is closest to the middle (n/2).
2. CD: Contradiction-detection block (Figure 6)
3. CP: Choice-point-selection Block
4. LB: Logic block
5. RAM: Backtrack memory
59
FIGURE 7. Inference block.
60
Experimental Mathematics, Vol. 17 (2008), No. 1
FIGURE 8. The choice-point-selection block.
The LB (logic block) takes the output of all three previously described blocks and does the following. If the assignment contains a contradiction (as would be indicated
by the CD block), then backtracking will be triggered,
and the previously saved assignment will be retrieved
from RAM to be expanded. If there is no assignment
in memory, the computation is done, and no assignment
without an arithmetic progression is possible.
If the assignment does not contain a contradiction but
a new coloring was inferred by the IB block, then the
inference is made and set as the new input to all three
previously described blocks to check for contradictions
and possible additional inferences.
Finally, if the assignment contains neither a contradiction nor new inferences, a new choice point is selected.
If all numbers are assigned a color, then the solution has
been found. Otherwise, the selected number, colored using the first color, is the current new assignment, and the
selected number, colored using the second color, is saved
into RAM for future backtracks.
This design simplifies our fastest SAT-solver version
by removing the delayed literal evaluation logic while still
maintaining excellent speedup over the sequential version
and taking advantage of the fine-grained parallelism in an
FPGA.
9.
PERFORMANCE RESULTS
For illustration, our implementation in C of the solver can
prove W (2, 5) = 178 in 2.613 seconds, provide a partition
of length 177 for W (2, 5) in 0.091 seconds, and provide all
satisfiable assignments for W (2, 5) for n = 177 in 2.683
seconds on an Intel Pentium 4 running at 3 GHz.
For the W (2, 6) search using parallel backtracking we
ran the solver on the following clusters:
• 66x AMD Opteron cores running at 1800 MHz;
• 34x AMD Athlon running at 1533 MHz;
• 64x Intel PIII running at 450 MHz;
• around 50 other Intel and Sun Sparc-based processors not exceeding 500 MHz each.
Given the above computational resources, it is our estimate that it would take more than a year to complete
the search of all 2,537,546 preprocessed patterns using
only cluster-based computation.
The wall processing time (time from start to finish)
was about 253 days on clusters that were time-shared
with other students and faculty. We had over 200 processors working on the problem at various times. The
first half of the search was done using time-shared clusters only (and took about six months), and the second
half was done by a combination of time-shared clusters
and four dedicated Xilinx Virtex-4 based FPGAs. The
estimated time for an FPGA-only computation is three
months, which could even be shortened with an improved
design or an increased number of FPGAs.
10.
EXTREME PARTITIONS
There are exactly 3552 extreme partitions for W (2, 6) of
length 1131. Each of these partitions involves variations
of the basic pattern B of length 56, where
B = 011110010000010111101110011111011011101000
11101001010011.
Letting A1 = BBrev,neg and A2 = Bneg Brev , half of these
3552 patterns are of the form
xA1 xA2 xA1 xA2 xA1 xA2 xA1 xA2 xA1 xA2 x,
where the 11 “glue”variables x can be assigned 0, 1 values independent of their position in the partition. The
only constraint on these glue variables is that the 0, 1 assignment to them not create an arithmetic progression of
length 6 (among themselves). One of these extreme partitions is illustrated in Figure 9. The other 1776 extreme
partitions for W (2, 6) are obtained by negating the above
1776 extreme partitions.
11.
CONCLUSION AND FUTURE WORK
Finding the value of van der Waerden numbers presents
a challenging problem, since the underlying principle behind their computation is still unknown. Using preprocessing resulted in a significant reduction of the search
space, which together with optimized SAT solvers and
(eventually) hardware support in the form of FPGAs allowed for the computation of the sixth known van der
Waerden number W (2, 6) = 1132.
We have tested the same approach with W (2, 7), and
the preprocessing does not reduce the size of the search
Kouril and Paul: The van der Waerden Number W (2, 6) Is 1132
12.
61
ACKNOWLEDGMENTS
This work has been supported by NSF Grants 9871345 and
0521189, and also by a fellowship grant to Michal Kouril by
the Ohio Board of Regents.
REFERENCES
[van der Waerden 27] B. L. van der Waerden. “Beweis einer
Baudetschen Vermutung.” Nieuw Archief voor Wiskunde
15 (1927), 212–216.
FIGURE 9. An extreme partition for W (2, 6).
space sufficiently to be computable at this time. We have
also applied the same preprocessing technique to computing multicolor van der Waerden numbers, and in this
case the results look promising. It may also be possible
to prove longer unavoidable patterns. We believe that
with more optimization and clever theoretical work more
numbers can be verified.
[Beeler and O’Neil 79] M. D. Beeler and P. E. O’Neil. “Some
New van der Waerden Numbers.” Discrete Mathematics
28 (1979), 135–146.
[Kouril and Franco 05] M. Kouril and J. Franco. “Resolution
Tunnels for Improved SAT Solver Performance.” In Eighth
International Conference on Theory and Applications of
Satisfiability Testing, St. Andrews, Scotland, pp. 143–157.
Berlin / Heidelberg: Springer, 2005
[Kouril 06] M. Kouril. “A Backtracking Framework for Beowulf Clusters with an Extension to Multi-cluster Computation and SAT Benchmark Problem Implementation.”
PhD thesis, University of Cincinnati, 2006.
Michal Kouril, Computer Science Department, 814B Rhodes Hall, University of Cincinnati, Cincinnati, OH 45221-0030
([email protected])
Jerome L. Paul, Computer Science Department, 814B Rhodes Hall, University of Cincinnati, Cincinnati, OH 45221-0030
([email protected])
Received May 5, 2007; accepted November 3, 2007.